Question

In a group of children each child gives a gift to every other child. If the number of gifts are 132, find the number of children.
A. 12
B. 18
C. 10
D. 16

Answer 1

Hint: In this problem the concept of quadratic equations will be used. First we will assume a suitable variable for the total number of children, that is n. Each child will give a total of (n - 1) gifts to all the other children. Using this information, we will form an equation in n, to get our final answer using a suitable method.

Complete step-by-step answer:

We have to find the number of children in the group. Let the number of children in the group be n. Each student gives one gift to all the other students, so each child will give a total of (n - 1) gifts. So, the total number of gifts will be the sum of gifts given by each student, and is equal to 132 as given in the question.
(n - 1) + (n - 1) + (n - 1) + …. n times = 132
Adding the same number n times means that we will multiply (n - 1) by n.
n(n - 1) = 132
Opening the brackets and transposing 132 to the LHS we get-
${{{n}}^2} - {{n}} = 132$
${{{n}}^2} - {{n}} - 132 = 0$
We will now use the factorization method to further simplify this equation. We can see that the sum of the factors here is -1, and their product is -132. We will identify a pair such that they satisfy both these conditions. This pair is -12 and 11, so we can proceed as-
${{{n}}^2} - 12{{n}} + 11{{n}} - 132 = 0$
$n(n - 12) + 11(n - 12) = 0$
$(n - 12)(n + 11) = 0$
$n = 12, -11$
We clearly know that the number of students cannot be negative. So the value of n is 12. The correct option is A.

Note: An alternative and lengthier method to solve the roots, that is n, is by using the quadratic formula. Another type of mistake is that students write the roots of the equation as -12 and 11, instead of 12 and -11. To avoid this, we should always substitute the root in the original equation to check the final answer.